The direction of the bird's velocity can be adjusted so that the wind and the resulting bird velocity will both be in the southerly direction.
(a) The bird should fly at an angle of roughly South 23.58° East if it wants to move directly southward in relation to the earth.
(b) The bird would need around 5.46 hours to fly 500 kilometers from North to South.
The components of the velocity are v(x) and v(y).
The magnitude of the velocity of the bird is 100 km/h
The orientation of the bird corresponds to that of North and South.
The magnitude of the velocity of the wind is 40 km/h
East-west is the direction of the wind.
(a) The component vector of the two velocities are;
The velocity of the bird, v(y) = -100(j)
The velocity of wind, v(x) = 40(i)
If the bird has a component of velocity to accommodate the wind velocity, the direction of its movement will be southerly.
Let θ be the angle of the direction of the bird relative to the negative x-axis, therefore:
-100(j) × cos(θ) = -40·i
θ = cos⁻¹(40/100)
θ ≈ 66.42° in the third quadrat, given that both the y, and x, values are negative.
In the South-West direction, the angle is:
90 - 66.42 ≈ 23.58°
Therefore, to fly directly southward relative to the ground, the bird should fly in the direction of south 23.58° east.
(b) The following is given as the component of the bird's southerly velocity when traveling in the new direction:
v(s) = 100 × sin(θ)
Hence,
Southward velocity, v(s) = 100 × sin(66.42°)
The time t it takes to travel 500 km, is given as:
d = v(s)t
500 = 100 × sin(66.42°)(t)
t = 5.46 hrs
The bird would need 5.46 hours to fly 500 kilometers from North to South.
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The complete question is mentioned below:
Canada geese migrate essentially along a north–south direction for well over a thousand kilometers in some cases, traveling at speeds up to about 100 km/h. If one goose is flying at 100 km/h relative to the air but a 40-km/h wind is blowing from west to east,
(a) at what angle relative to the north–south direction should this bird head to travel directly southward relative to the ground?
(b) How long will it take the goose to cover a ground distance of 500 km from north to south? (Even on cloudy nights, many birds can navigate by using the earth’s magnetic field to fix the north–south direction.)
